Owners of some U.S. natural gas storage sites began turning some customers away this week for fear that the system will overflow in autumn, the clearest sign yet that surging production may wreak havoc in the market later this year.

Huge supplies of shale gas from newly exploited deposits are set to fill U.S. storage caverns and tanks to their limit this year for the first time, threatening to force producers to shut down wells or risk overloading the pipeline network…